I will discuss two classical problems in analysis and geometry posed
on n-dimensional flat torus. I will focus on some interesting
questions which remain open. I will then discuss a nonlocal
isoperimetric problem with an emphasis on the calculation and
application of its first and second variations.

This talk is based on joint work with Peter Sternberg (Indiana
University).

In this talk, we consider the problem of minimizing the energy
functional ò(|Ñu|2 + c{u > 0}). We will exhibit
the first example of a singular energy minimizer, which occurs in
dimension n=7. This is the analogue of the 8-dimensional Simons
cone in the theory of minimal surfaces.

In control theory one traditionally uses an exponential discount
factor to arbitrage future gains agains current losses. It is well
known that the optimal control problem then is solved by a
Hamilton-Jacobi-Bellman equation for the value function. In joint
work with Ali Lazrak, we point out that this approach completely
breaks down when the discount is not exponential, and the
decision-maker cannot commit. Instead of looking for optimal
controls, one must then look for equilibrium strategies, and the HJB
equation is then replaced by a remarkable integro-differential
equation.

I will discuss some recent progress on the semilinear elliptic problem
Du = [(lf(x))/((1+u)2)] on a smooth bounded domain
W of RN with an homogeneous Dirichlet boundary
condition. This equation models a simple electrostatic
Micro-Electromechanical System (MEMS) device and has been studied
recently by Pelesco, and by Guo-Pan-Ward. Guo and Ghoussoub
show-among other things-that the branch of minimal solutions
ul is compact up to a certain critical value l*,
provided 1 £N£ 7. In this talk, I will describe an analogous
result for the second branch (of "mountain pass" solutions), which
holds in the same low dimensions. Our techniques rely on a careful
blow-up analysis for solutions satisfying certain spectral properties.

We give a new proof of Klein's result on the existence of absolutely
continuous spectrum for a discrete random Schrödinger operator on a
tree with small disorder. Our proof relies on a new geometric way of
controlling the Green's function, based on the contraction properties
of a transformation in hyperbolic space.

We establish existence and multiplicity of solutions for the Dirichlet
problem åi¶iiu + [(|u|2*-2u)/(|x|s)] = 0 on
smooth bounded domains W of Rn (n³ 3) involving the
critical Hardy-Sobolev exponent 2*
= [(2(n-s))/(n-2)] where
0 < s < 2, and in the case where zero (the point of singularity) is on
the boundary ¶W. Just as in the Yamabe-type
non-singular framework (i.e., when s=0), there is no
nontrivial solution under global convexity assumption (e.g.,
when W is star-shaped around 0). However, in contrast to the
non-satisfactory situation of the non-singular case, we show the
existence of an infinite number of solutions under an assumption of
local strict concavity of ¶W at 0 in at least one
direction. More precisely, we need the principal curvatures of
¶W at 0 to be nonpositive but not all vanishing. We
also show that the best constant in the Hardy-Sobolev inequality is
attained as long as the mean curvature of ¶W at 0 is
negative. The key ingredients in our proof are refined concentration
estimates which yield compactness for certain Palais-Smale sequences
which do not hold in the non-singular case.

The Schroedinger map equation is a basic model in ferromagnetism, as
well as a geometric (and hence non-linear) version of the linear
Schroedinger equation. An important open question is whether finite
energy solutions are globally smooth, or blow up in finite time. We
describe some results for equivariant Schroedinger maps from
2+1-dimensional space-time into the 2-sphere, with energy close to
the energy of harmonic maps.

Regularity properties of Aleksandrov solutions to the Dirichlet
problem for the Monge-Ampère equation detD2u = m where
m is a Borel measure on a convex domain in Rn will be
discussed. The measure m satisfies a condition, introduced by
Jerison, that is weaker than the doubling condition. Some of the
results of Caffarelli's regularity theory for the Monge-Ampère
equation, more specifically strict convexity and interior
C1,a regularity, are extended to the solutions of these
problems.

We study a simple model of crystalline surfaces in R3.
Microscopically, these are random discrete surfaces, arising in the
so-called dimer model, or domino tiling model. The law of large
numbers implies that at large scales the surfaces take on definite
shapes, which are smooth surfaces satisfying a certain PDE, related to
the complex Burgers equation. We show how this equation can be solved
via complex analytic functions, and investigate the behavior of
solutions, in particular the formation of facets. This is the first
model of facet formation which can be analytically solved.

In an new attempt to model the phase transition from nucleate to
transient boiling, Professor Marquardt from Aachen proposed to
consider the the heat flow in the heating vessel rather than in the
boiling liquid. In this model the heat flow in the wall of the
vessel, subject to heat equation, is combined with a nonlinear Neumann
boundary condition at the surface of the wall toward the boiling
liquid. The nonlinearity is determined by the change of the heat
conduction coefficient in the phase transition.

For the one-dimensional heat equation with a nonlinear inhomogeneous
term, the existence of wavefront solutions is well known and widely
used to model phase transitions. Work of Aronson and Weinberger also
dealt with the more-dimensional situation and showed that there are
sub-solutions which behave like wavefronts. Consequently the actual
solutions must have a sudden change of state, also. However, this
model with the nonlinearity in the equation rather then the boundary
condition can be justified in our case for (infinitely) thin surfaces
only.

Here we present an approach which provides a wavefront type
sub-solution for the nonlinear Neumann problem and hence establish a
first mathematical confirmation of Marquardt's model.

Further research will concentrate on the discussion of initial
configurations (dry spots) which generate a wavefront type solutions
and those which do not. In addition we intend to address the question
how the maximum and minimum speed of the traveling wave is determined
by the initial configuration and the other parameters of the data.

Self-dual variational principles are introduced in order to construct
solutions for Hamiltonian and other dynamical systems which satisfy a
variety of linear and non-linear boundary conditions including many of
the standard ones. These principles lead to new variational proofs of
the existence of parabolic flows with prescribed initial conditions,
as well as periodic, anti-periodic and skew-periodic orbits of
Hamiltonian systems.

In this talk, we will discuss Monge-Ampère type equations arising
in optimal transportation problems. We prove the comparison principle,
maximum principle and also a quantitative estimate of Aleksandrov type
for c-convex functions. These results are in turn used to prove the
solvability and uniqueness of weak solutions for the Dirichlet
problems.

Numerical results for the mass tranportation problem will be
presented. The transportation problem with cost function which depend
on the difference x-y will be considered. We approximate measures
by atoms, and project to a finite dimensional linear programming
problem, which can then be solved by standard methods.

Numerical results for linear, quadratic, and square root of distance
costs will be presented.

We will also present a convergent finite difference method for solving
the Dirichlet problem for the Monge-Ampère equation.

We will consider in this talk the inverse problem of determining the
electrical conductivity inside a medium by making voltage and current
measurements on subsets of the boundary. We will state and outline
the proofs of some recent results on this problem.

An optimization problem for the fundamental eigenvalue of the
Laplacian in a planar simply-connected domain that contains N small
identically-shaped holes, each of a small radius e << 1, is
considered. The boundary condition on the domain is assumed to be of
Neumann type, and a Dirichlet condition is imposed on the boundary of
each of the holes. The reciprocal of this eigenvalue is proportional
to the expected lifetime for Brownian motion in a domain with a
reflecting boundary that contains N small traps. For small hole
radii e, we derive an asymptotic expansion for this
eigenvalue in terms of the hole locations and the Neumann Green's
function for the Laplacian. For the unit disk, ring-type
configurations of holes are constructed to optimize the eigenvalue
with respect to the hole locations. For a one-hole configuration, the
uniqueness of the optimizing hole location in symmetric and asymmetric
dumbbell-shaped domains is investigated. This eigenvalue optimization
problem is shown to be closely related to the problem of determining
certain vortex configurations in the Ginzburg-Landau theory of
superconuctivity and to the problem of determining equilibrium
locations of particle-like solutions, called spikes, to certain
singularly perturbed nonlinear reaction-diffusion systems. Some
results for the equilibria, stability, and bifurcation behavior, of
these spike solutions are given.